The theory of combinatorial limits is an evolving area of combinatorics.
The most developed is the theory of graph limits,
which is covered in detail in a recent monograph by Lovász [1]}.
Further results concerning many other combinatorial structures exist,
e.g. for permutations [2]}, [3]}, [4]}, [5]} or for partial orders [6]}, [7]}.
In the case of graphs, limits of dense graphs [8]}, [9]}, [10]}, [11]}, [12]},
also see [13]}, [14]} for a general theory of limits of dense combinatorial structures, and
limits of sparse graphs [15]}, [16]}, [17]}, [18]}, [19]}, [20]} evolved to a large extent independently.
A notion of first order convergence was introduced by Nešetřil and Ossona de Mendez [21]}, [22]}
as an attempt to unify convergence notions in the dense and sparse regimes.
This general notion can be applied in the setting of any relational structures,
see e.g. [23]} for results on limits of mappings.
Informally speaking,
a sequence of relational structures is first order convergent
if for any first order property, the density of \(\ell \) -tuples of the elements having this property converges;
a formal definition is given in Subsection REF .
Every first order convergent sequence of dense graphs
is convergent in the sense of dense graph convergence from [9]}, [10]}, and
every first order convergent sequence of graphs with bounded degree
is convergent in the sense of Benjamini-Schramm convergence as defined in [16]}.
